Prove that there is no permutation α such that α(12) α^-1 = (124)(567)

Well (1 2 3) = (1 2)(2 3), which is an even permutation.
So must be even, but this is impossible because if is an even permutation, then is odd, so is odd.
But if is odd, then is even, so is odd. So the left side is never an even permutation.